Author: Aggeliki Papadaki

Σας προσκαλούμε στην ομιλία του κ. Jasper Rou (ΥΔ Delft University of Technology, http://www.jasperrou.nl/), η οποία θα πραγματοποιηθεί  την Παρασκευή,  14 Ιουνίου 2024 & ώρα 13:00, στην αίθουσα Σεμιναρίων του Τομέα Μαθηματικών.

Title: Convergence of time-stepping Deep Gradient Flow Methods

Abstract: In this research, we consider the convergence of neural network algorithms for partial differential equations (PDE). More specifically, we consider a Time-stepping Deep Gradient Flow method, where the PDE is solved by discretizing it in time and writing it as the solution of minimizing a variational problem. A neural network approximation is then trained to solve this minimization using stochastic gradient descent. This method reduces the training time compared to for instance the Deep Galerkin Method. We prove two things.

First, that there exists a neural network converging to the solution of the PDE. This proof consists of three parts: 1) convergence of the time stepping; 2) equivalence of the solution of the discretized PDE and the minimizer of the variational formulation and 3) the approximation of the minimizer by a neural network by using a version of the universal approximation theorem. Second, we prove that when training the network we converge to the correct solution. This proof consists of two parts: 1) as the number of neurons goes to infinity we converge to some gradient flow and 2) as the training time goes to infinity this gradient flow converges to the solution.

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